Fiber optic shape sensor

ABSTRACT

A shape sensing system to determine the position and orientation of one link with respect to another link in a kinematic chain. An optical fiber is coupled to two or more links in a kinematic chain. A shape sensing segment is defined to start at a proximal link and to end at a distal link, crossing one or more joints. A reference frame is defined at the start of the shape sensing segment. As the joints move, an interrogator senses strain in the shape sensing segment. The sensed strain is used to output a Cartesian position and orientation of the end of the shape sensing segment with respect to the reference frame defined at the start of the shape sensing segment. The pose of the kinematic chain is determined from the Cartesian positions and orientations of one or more shape sensing segments defined for the kinematic chain and from an a priori model and constraints of the kinematic chain.

BACKGROUND

1. Field of Invention

Inventive aspects are associated with shape sensing, more particularlyto sensing position and orientation at various points of a kinematicchain, and still more particularly to sensing the shape of a flexiblebody and sensing the position and orientation of an end effectorcomponent of a surgical instrument or of an entry guide for one or moresurgical instruments in a telerobotic surgical system.

2. Background Art

FIG. 1 is a diagrammatic view of an optical fiber core portion 100. Thesurrounding cladding and fiber are omitted for clarity. Two fiber Bragggratings (FBG's) 102 a, 102 b are shown formed in fiber core portion100, which are illustrative of many such FBG's typically formed alongthe full length of a core. The many vertical lines shown in each FBG 102represent the changes in refractive index that characterize an FBG. Asshown in FIG. 1, the FBG's 102 a, 102 b are separated by a tethersegment 104, which is completely transmissive.

As is known, each of the FBG's 102 may be interrogated for straininformation. In a fiber that contains two or more cores, with FBG's ineach core positioned at the same location along the fiber, the fiber'sbend direction and amount of axial twist may be determined from thestrains in each core's FBG's. From the strain information from each coreat each FBG location, and from the known tether segment length (e.g., 5mm, 1 cm), the position of the location with the next set of FBG's canbe estimated. In this way, the fiber shape associated with theinterrogated FBG's may be determined. U.S. Pat. App. Publ. No. US2006/0013523 A1 (filed 13 Jul. 2005), which is incorporated herein byreference, discloses a fiber optic position shape sensing device andmethod. In one instance, strain information is used to determine thebend angle between two links, as described in U.S. Pat. App. Publ. No.US 2007/0156019 A1 (filed 20 Jul. 2006), which is incorporated herein byreference. Such bend information may be used in forward kinematiccalculations to determine the position of the distal end of a kinematicchain (a set of links coupled by one or more movable mechanicalconstraints). One non-limiting example of such a kinematic chain is aminimally invasive surgical instrument with one or more revolute joints.

In one instance, a curvilinear coordinate system is defined with anorigin at the connection where the fiber is joined to the interrogatorunit (i.e., at the proximal end of the fiber). In addition, a Cartesianframe is also defined as a base reference frame having an origincoincident with the curvilinear coordinate system's origin. Using knowntechniques, each of the FBG's 102 are interrogated for straininformation.

Known techniques to interrogate the FBG's by processing the lightreflected back when an optical light source is coupled with the fiberare described in, for instance, Andreas Othonos & Kyriacos Kalli,Fundamentals and Applications in Telecommunications and Sensing Ch. 7,301-388 (Arthech House 1999), which is incorporated herein by reference.Such interrogation techniques include on the use of edge filters,tunable filters, interferometers, tunable lasers, and CCD spectrometers.Each technique provides different spatial resolution in the measurementof strain along a fiber core, different speeds of interrogation, i.e.,update of the measurement, and different levels of immunity todisturbances to the strain measurements such as those produced byvariation of temperature, light polarization, source light variations,and losses of light along the fiber. Among the interferometrictechniques that are based on detection of the phase of the reflectedlight are Optical Time Domain Reflectometry and Optical Frequency DomainReflectometry (OFDR), as described in U.S. Pat. No. 5,798,521 (filed 27Feb. 1997) and in U.S. Pat. App. Publ. No. US 2007/0065077 A1 (filed 26Sep. 2006), which are incorporated herein by reference.

To determine the fiber's approximate shape, the strain informationmeasured at each FBG location is used to determine the approximate localbend for the length of fiber without FBG's (e.g., over a 1 cm tethersegment). For example, the strain information from three cores in afiber is used to compute the plane and the bend radius of the fiber.Segments are defined at various locations along the fiber, and eachsegment ends at a co-located ring of FBG's in the three cores. Given theCartesian x,y,z position of the FBG ring being processed (i.e., thesegment end position), the position of the next FBG ring can be computedwith simple geometry. The position of the first segment's end locationwith respect to the base frame is then determined from the firstsegment's bend information. Next, strain information for the secondsegment is processed to determine the second segment's bend. The secondsegment's bend information is combined with the position of the firstsegment's end location to determine the second segment's end locationposition with respect to the base frame. Thus the position of eachsegment end location is determined with respect to the base frame, andthe position information is used to determine the approximate shape ofthe fiber.

There are, however, disadvantages to current optical fiber shape sensingmethods. To begin with, such methods are based on the average strainmeasured in each FBG, and so the FBG size limits the measurementresolution. In addition, the shape (or state) of the fiber isreconstructed as a vector of equally spaced three-dimensional (3D)points in world coordinates. The result is a large data set that getslarger as measurement resolution is increased. Also, the resolution ofthe 3D points is limited by the spacing between FBG's. Further, in thesemethods there is an assumption that the length of fiber between measuredFBG's has a constant bend radius, and this assumption can reduce theaccuracy of the sensed shape as compared with the fiber's actual shape.Another disadvantage is that the computation of the fiber tangent vectorat a particular fiber location (i.e., the direction the fiber ispointing at that location) requires differentiation of the 3D points.This differentiation delivers even lower measurement resolution. Yetanother disadvantage is that the 3D point data set is not in a form thatis required for the type of processing needed when the fiber is embeddedinto a kinematic chain, and therefore the kinematic chain's pose has tobe inferred from the sensed fiber position. This limitation isespecially true if the fiber is embedded in a kinematic chain but isallowed to slide with reference to one or more links. In suchconfigurations, the fiber may not follow a path that exactly correspondsto a bend in one or more of the kinematic chain's joints, and frictionbetween the fiber's surface and a surrounding conduit through one ormore links may influence the fiber's shape.

What is needed, therefore, is a more effective and accurate way todetermine the shape of an optical fiber and a more effective way ofproducing the shape information for use in determining the position andorientation of all the links of a kinematic chain, i.e., the pose of akinematic chain, that is associated with the fiber. These needs areespecially true for various real time implementations, such as fortelerobotically controlled minimally invasive surgical instruments.

SUMMARY

A shape sensing system to determine the position and orientation of onelink with respect to another link in a kinematic chain is described. Anoptical fiber is coupled to two or more links in a kinematic chain. Ashape sensing segment is defined to start at a proximal link and to endat a distal link, crossing one or more joints. A reference frame isdefined at the start of the shape sensing segment. The reference framemay be defined in various ways, and one way is to define one axis normalto the fiber and another axis tangent to the fiber.

As the joints move, an interrogator senses strain in the shape sensingsegment. The sensed strain is used to output a Cartesian position andorientation of the end of the shape sensing segment with respect to thereference frame defined at the start of the shape sensing segment. TheCartesian position and orientation may be used to define a referenceframe for a subsequent shape sensing segment for one or more additionallinks and joints in the kinematic chain. The Cartesian position andorientation is determined and output in a format that facilitatescalculation. The pose of the kinematic chain is determined from theCartesian positions and orientations of one or more shape sensingsegments defined for the kinematic chain, from an a priori model of thekinematic chain, and from the geometrical constraints between the shapesensing fiber and the kinematic chain.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagrammatic view of an optical fiber core portion.

FIG. 2 is a diagrammatic view of another optical fiber core portion.

FIGS. 3A-3E are diagrammatic views of optical fiber cores.

FIGS. 4A and 4B are diagrammatic views of the proximal ends of opticalfibers with an illustrative reference frame.

FIG. 5 is a diagrammatic view that illustrates reference frames atsegment start locations in an optical fiber used for shape sensing.

FIGS. 6A and 6B are diagrammatic views that illustrate determiningsegment end position and orientation.

FIGS. 7A, 7B, and 7C are diagrammatic views of an illustrative kinematicchain that includes two elongate, rigid mechanical links interconnectedby one or more mechanical constraints.

FIG. 8 is a schematic view of a minimally invasive surgical instrument.

FIG. 9 is a schematic view of another minimally invasive surgicalinstrument. FIG. 9A is an illustrative schematic of parallel motionmechanism in a displaced pose, and FIGS. 9B and 9C are elevation viewsof an illustrative embodiment showing the parallel motion mechanism inline with a shaft of the surgical instrument (FIG. 9B) and displaced inposition (FIG. 9C).

FIG. 10 is a diagrammatic view of a kinematic chain that includes twolinks interconnected by a revolute joint.

FIG. 11 is a schematic view of a kinematic chain that is illustrative ofkinematic chains made of two or more flexible mechanical sections.

FIG. 12 is a diagrammatic view of a shape sensing system.

DETAILED DESCRIPTION

This description and the accompanying drawings that illustrate aspectsand embodiments of the present invention should not be taken aslimiting—the claims define the protected invention. Various mechanical,compositional, structural, electrical, and operational changes may bemade without departing from the spirit and scope of this description andthe claims. In some instances, well-known circuits, structures, andtechniques have not been shown or described in detail in order not toobscure the invention. Like numbers in two or more figures represent thesame or similar elements.

Further, this description's terminology is not intended to limit theinvention. For example, spatially relative terms—such as “beneath”,“below”, “lower”, “above”, “upper”, “proximal”, “distal”, and thelike—may be used to describe one element's or feature's relationship toanother element or feature as illustrated in the figures. Thesespatially relative terms are intended to encompass different positionsand orientations of the device in use or operation in addition to theposition and orientation shown in the figures. For example, if thedevice in the figures is turned over, elements described as “below” or“beneath” other elements or features would then be “above” or “over” theother elements or features. Thus, the exemplary term “below” canencompass both positions and orientations of above and below. The devicemay be otherwise oriented (rotated 90 degrees or at other orientations)and the spatially relative descriptors used herein interpretedaccordingly. Likewise, descriptions of movement along and around variousaxes includes various special device positions and orientations. Inaddition, the singular forms “a”, “an”, and “the” are intended toinclude the plural forms as well, unless the context indicatesotherwise. And, the terms “comprises” “comprising”, “includes”, and thelike specify the presence of stated features, steps, operations,elements, and/or components but do not preclude the presence or additionof one or more other features, steps, operations, elements, components,and/or groups. Components described as coupled may be electrically ormechanically directly coupled, or they may be indirectly coupled via oneor more intermediate components. All examples and illustrativereferences are non-limiting and should not be used to limit the claimsto specific implementations and embodiments described herein and theirequivalents. The headings are solely for formatting and should not beused to limit the subject matter in any way, because text under oneheading may cross reference or apply to text under one or more headings.

The term “flexible” in association with a mechanical structure orcomponent should be broadly construed. In essence, it means thestructure or component can be bent without harm. For example, a flexiblemechanical structure may include a series of closely spaced componentsthat are similar to “vertebrae” in a snake-like arrangement. In such anarrangement, each component is a short link in a kinematic chain, andmovable mechanical constraints (e.g., pin hinge, cup and ball, and thelike) between each short link may allow one (e.g., pitch) or two (e.g.,pitch and yaw) degrees of freedom (DOF) of relative movement between thelinks. As another non-limiting example, a flexible mechanical structuremay be continuous, such as a closed, bendable tube (e.g., Nitinol,polymer, and the like) or other bendable piece (e.g., kerf-cut tube,helical coil, and the like). Accordingly, a short, flexible structuremay serve as, and be modeled as, a single mechanical constraint (joint)providing one or more DOF's between two links in a kinematic chain, eventhough the flexible structure itself may be a kinematic chain made ofseveral coupled links.

I. Fiber Optic Shape Sensor

FIG. 2 is a diagrammatic view of another optical fiber core portion 200.The surrounding cladding and fiber are omitted for clarity. Core portion200 shows illustrative FBG's 202 a-202 e formed adjacent one another inthe optical fiber core. There is no tether segment between the FBG's. Insome instances a boundary region, illustrated by regions 204 a-204 d,separates adjacent FBG's such that the spacing between the refractiveindex changes in each FBG 202 a-202 e may not be exactly the same as thespacing between FBG's. Again, the many vertical lines shown in each FBG202 represent the changes in refractive index that characterize an FBG.

A distinctive feature of using OFDR, and interferometric techniques ingeneral, for strain interrogation with a fiber as illustrated by FIG. 2is that it allows sensing the strain in the fiber over the FBG's with avery high resolution—on the order of microns for fibers of 10 mlength—as determined by the interrogating laser source coherence length.

FIGS. 3A-3E are diagrammatic views of various configurations of opticalfiber cores. The surrounding cladding and fiber are omitted for clarity.Each of the many vertical lines shown in the illustrative coresrepresent individual, adjacent FBG's (e.g., FBG's 202 a-202 e as shownin FIG. 2). In one embodiment, approximately 60 FBG's are formed forevery 20 mm length of a core/fiber. As described below, each core is oneof three or more cores in a single fiber.

As shown in FIG. 3A, a curvilinear coordinate system s(t) is defined forthe fiber, and hence for core 300 and the other cores (not shown). Insome cases the location L₀ of the origin of s(t) is defined at theproximal end 302 of the fiber, where the fiber connects to a straininformation interrogator unit. In other cases, the origin location L₀ ofs(t) may be defined at a location 304 along the fiber. For example, L₀may be defined at a location within a base mechanical link of akinematic chain at which the fiber is fixed during manufacturing.

Then, once the origin location is defined, one or more shape sensingsegments (bend segments) are defined between locations along the core.Each defined shape sensing segment of a core contains part of one FBG,or one full FBG, or many adjacent FBG's. As shown in FIG. 3A, forexample, three shape sensing segments 306 a-306 c are defined. Eachshape sensing segment is defined in s(t) as beginning at a segment startlocation on the fiber and continuing along for a particular segmentlength until a segment end location on the fiber. As shown in FIG. 3A,for example, segment 306 a begins at a first location L₁ and continuesfor length s₁. Similarly, segment 306 b begins at segment start locationL₂ and continues for length s₂, and segment 306 c begins at segmentstart location L₃ and continues for length s₃. Segments of equal length(e.g., segments 306 b, 306 c as shown) or of different lengths (e.g.,segments 306 a, 302 b as shown) may be defined. In addition, the shapesensing segments need not be adjacent one another—they can be defined tooverlap. And, the most distal shape sensing segment may or may not bedefined to end at the distal end of the fiber.

FIG. 3B illustrates a fiber core similar to the core illustrated in FIG.3A, but which has segments 308 a, 308 b of adjacent FBG's separated by afiber length 310 without FBG's. This configuration differs from the oneillustrated in FIG. 1, in which each FBG is separated by a tethersegment. A configuration as illustrated by FIG. 3B may be used, forexample, in a kinematic chain with long, straight links coupled byrevolute joints. Each segment 308 a, 308 b is positioned to correspondto a joint and is used to sense fiber strain associated with thecorresponding joint's movement. The length 310 is positioned tocorrespond to a long, rigid portion of a link (see e.g., the linkillustrated in FIGS. 9A-9C below). Since the long, rigid link neverbends, sensing the fiber's shape within the link is not required. Thesegments 308 a, 308 b are used to sense the relative position andorientation between adjacent links. The most distal FBG segment may beplaced at the most distal end of the fiber, or a fiber length withoutFBG's may exist between the most distal FBG segment and the fiber'sdistal end.

FIG. 3C also illustrates a fiber core similar to the core illustrated inFIG. 3A, but which has only a single segment 312 of adjacent FBG's atthe fiber's distal end. A configuration as illustrated by FIG. 3C may beused, as a non-limiting example, to sense only movement of a distalportion of a kinematic chain (e.g., only the position and/or orientationof an end effector at the distal end of a minimally invasive surgicalinstrument, and/or the position of a jaw in a jawed end effector(grasper, scissors, and the like)).

FIGS. 3D and 3E are similar to FIGS. 3B and 3C, but the cores in FIGS.3D and 3E have FBG's defined along their entire length, althoughsegments are defined in non-adjacent FBG portions of the core. As shownin FIG. 3D, for example, segments 314 a and 314 b are defined with alength 316 of FGB-configured core between them. As shown in FIG. 3E,segment 318 is defined at the distal end of the core, with no additionalsegments defined in the FBG-configured core.

FIG. 4A is a diagrammatic view of the proximal end of an optical fiber400 with an illustrative reference frame defined. As shown in FIG. 4A,fiber 400 has three FBG-configured cores 402 a-402 c within a claddinglayer 404. Each core 402 a-402 c is positioned at an apex of anequilateral triangle centered in the fiber.

As shown in FIG. 4A, a Cartesian reference frame is defined for thefiber 400. One axis of the Cartesian reference frame intersects one ofthe cores (the x-axis is shown intersecting core 402 a as anillustration) and another axis is tangent to the fiber 400's centerline(the z-axis is shown as an illustration). Defining the x-axis to extendthrough a core provides a rotational reference around core 400'scenterline. The definition of the x-axis is arbitrary and can be basedon the geometry of the kinematic chain embedding the fiber. Forinstance, the x-axis could be aligned to one joint axis of the kinematicchain in which the fiber is embedded or associated.

The Cartesian reference frame (x, y, z) shown in FIG. 4A functions as abase frame when defined with an origin coincident with the origin of thecurvilinear coordinate system s(t). When a Cartesian reference frame isdefined with an origin at a segment start location, it functions as ashape sensing segment reference frame. A Cartesian reference frame maybe similarly defined at a segment end location.

Although three cores are shown in FIG. 4A, other numbers of cores may beused (e.g., two opposite cores for planar bend measurement, four cores,etc.). FIG. 4B is a diagrammatic view of the proximal end of anotheroptical fiber 406. As shown in FIG. 4B, fiber 406 has fourFBG-configured cores 408 a-408 d within a cladding layer. Each core 408a-408 c is positioned at an apex of an equilateral triangle centered inthe fiber, and core 408 d is positioned at the center of the equilateraltriangle, which is coincident with the center of the fiber. Once again,an x-axis is defined intersecting a core (408 a) at an apex. The z-axisoriginates tangent to the core 408 d at the center of the triangle. Thecenter core 408 d may be used to enable sensing the angle of torsionaround the fiber's center longitudinal axis (strain between cores at theapexes, which varies with torsion of the fiber, is compared to strain inthe core at the center, which does not vary with torsion of the fiber).

In comparison, fibers without such a center core (e.g., the three-corefiber illustrated in FIG. 4A) cannot be used to sense the fiber's angleof torsion. For this reason, fibers without a center core are used inmechanical configurations that minimize torsional loads. In suchconfigurations, since the fiber has an inherent degree of torsionalstiffness, the x-axes of each reference frame defined along the fiber'slength maintain an essentially constant rotational relationship to eachother around the fiber's centerline.

FIG. 5 is a diagrammatic view that illustrates reference frames atsegment starts in an optical fiber used for shape sensing. FIG. 5depicts an optical fiber 500 that is, in one embodiment, configured withthree cores as illustrated in FIG. 4A (four cores as illustrated above,or other core configurations, may be used). In this illustrativeexample, each core is configured as shown in FIG. 3A, although variousFBG configurations may be used, as described above. Two shape sensingsegments are defined in fiber 500. The first segment 502 a is definedfrom curvilinear reference location L₁ (segment start) to curvilinearreference location L₁+s₁ (segment end). The second segment 502 b isdefined from curvilinear reference location L₂ (segment start) tocurvilinear reference location L₂+s₂ (segment end). In accordance withan aspect of the invention, a first Cartesian reference frame 504 a isdefined at segment start L₁. Reference frame 504 a's z-axis is tangentto the centerline of fiber 500 at segment start L₁, and reference frame504 a's x-axis runs through one of the cores as illustratively shown anddescribed in FIG. 4A. Similarly, a second Cartesian reference frame 504b is defined at segment start L₂, with reference frame 504 b's z-axistangent to the centerline of fiber 500 at segment start L₂, andreference frame 504 b's x-axis running through the same core asreference frame 504 a's x-axis.

The base reference frame illustrated in FIG. 4A and the two segmentstart reference frames illustrated in FIG. 5 are interrelated becauseall three have one normal axis (e.g., the x-axis) defined through thesame core (e.g., core 402 a).

In accordance with aspects of the invention, the position andorientation of each shape sensing segment end is determined with respectto the reference frame defined at the corresponding segment start. FIGS.6A and 6B are diagrammatic views that illustrate determining segment endposition and orientation. In FIG. 6A, an illustrative shape sensingsegment 600 of a fiber is shown. Segment 600 begins at segment start L₁and ends at segment end L₁+s₁. A Cartesian segment start reference frameis defined at segment start L₁ as described above. The position of thesegment end is identified by the distance change along each of thereference frame's axes; i.e., the position of the segment end withrespect to the segment start reference frame is Δx, Δy, Δz as shown inFIG. 6A. The orientation of the segment end is identified by tangentunit vector {right arrow over (t)} as depicted, whose three scalarcomponents are the three direction cosines t_(x), t_(y), t_(z) withrespect to the Cartesian reference frame. In FIG. 6A, segment 600 isshown with a simple, planar bend. But position and orientation may bedetermined for fiber segments having multiple bends within the threedimensions of the reference frame.

FIG. 6B shows another illustrative shape sensing segment 610 that hasmultiple bends in multiple planes within the segment start referenceframe. Segment 610 begins at segment start L₁ and ends at segment endL₁+s₁. A segment start reference frame is defined at segment start L₁ asdescribed above. And, as described with reference to FIG. 6A, theposition and orientation of segment end L₁+s₁ are identified as Δx, Δy,Δz, and {right arrow over (t)}, as shown in the drawing.

Thus, in accordance with aspects of the invention, strain informationfor each shape sensing segment is used to determine the position andorientation of the segment end independently of the path the fiber takesbetween the segment start and the segment end. This fiber pathindependence is used to an advantage for sensing the relative positionsand orientations of links in a kinematic chain, e.g., a robot arm withone or more actively controlled (e.g., servomotor actuated) joints.

A. Computation Example for Three Cores

The following is an illustration of computations carried out by anelectronic data processing unit. Skilled individuals will understandthat many hardware, firmware, and software options exist forconstructing an electronic data processing unit, and that implementationof necessary computations will be routine in light of this description.

The expression for the local strains(s) is written as a function ofdistance along a given fiber core,

ε_(n)=ε(Δdn)   (1)

where Δd is the distance increment per index n. The Δd value is set bythe resolution of the OFDR-based interrogator. For instance the localstrain ε(s) as a function of distance along each fiber core is obtainedby making use of an “Optical Backscatter Reflectometer”, a commerciallyavailable product from Luna Innovations Incorporated, Roanoke, Va., foreach core. Such a device is able to output the phase derivative of thereflected light as a function of the distance along the fiber core, asshown in Optical Backscatter Reflectometer User Guide Chaps 5-6, 33-60(Luna Technologies, Inc. 2004) (Document version 1.0 for OBR controlsoftware version 0.42 Beta), which is incorporated herein by reference.Such Phase Derivative information is proportional to the desired localstrain ε(s) in (1).

For bend calculations, the differential strains between the cores areneeded. For three cores, the required differential strains are:

Δε_(p,n)=ε_(2,n)−ε_(1,n)   (2a)

Δε_(q,n)=ε_(3,n)−ε_(1,n)   (2b)

where Δε_(p) and Δε_(q) designate the two differential strain arrays.These differential strains can then be converted into local bends in anortho-normal coordinate system by using a simple linear transformation,

$\begin{matrix}{\begin{bmatrix}\theta_{x,n} \\\theta_{y,n}\end{bmatrix} = {\begin{bmatrix}m_{px} & m_{qx} \\m_{py} & m_{qy}\end{bmatrix}\begin{bmatrix}{\Delta \; ɛ_{p,n}} \\{\Delta \; ɛ_{q,n}}\end{bmatrix}}} & \left( {3\; a} \right)\end{matrix}$

The m-matrix m is a fill description of the multi-core fiber, capturingthe effects of the locations of the cores and the initial rotationalorientation of the fiber in the coordinate system.

Next, these two rotation values are used to create a rotation matrixequal to the product of a first rotation of an angle θ_(x,n) around thex-axis and a second rotation of θ_(y,n) around the y-axis according tothe equations:

$\begin{matrix}\begin{matrix}{{\overset{\overset{\_}{\_}}{R}}_{x,n} = \begin{bmatrix}1 & 0 & 0 \\0 & {\cos \; \theta} & {{- \sin}\; \theta} \\0 & {\sin \; \theta} & {\cos \; \theta}\end{bmatrix}} \\{{\overset{\overset{\_}{\_}}{R}}_{y,n} = \begin{bmatrix}{\cos \; \theta} & 0 & {\sin \; \theta} \\0 & 1 & 0 \\{{- \sin}\; \theta} & 0 & {\cos \; \theta}\end{bmatrix}} \\{{\overset{\overset{\_}{\_}}{R}}_{n} = {{\overset{\overset{\_}{\_}}{R}}_{x,n}{\overset{\overset{\_}{\_}}{R}}_{y,n}}}\end{matrix} & \left( {4\; a} \right)\end{matrix}$

For small angle approximation, the above expression simplifies to:

${\overset{\overset{\_}{\_}}{R}}_{n} = \begin{bmatrix}1 & 0 & \theta_{x,n} \\0 & 1 & \theta_{y,n} \\{- \theta_{x,n}} & {- \theta_{y,n}} & 1\end{bmatrix}$

(5a) where, because a first order small angle approximation is used,R_(n) is a valid rotation matrix only if θ_(x)<<1 and θ_(y)<<1.

If sufficiently small spatial increments are used, the above conditionsare not difficult to satisfy. This rotation matrix is then moved intothe coordinate system at the n^(th) position on the fiber. In this way,the calculations are iterated to walk down the length of the fiber,reconstructing the tangent vector, as well as the vectors defining therotational coordinate system, along the way. The iterative equation is,

C _(n+1) = C _(n) R _(n)   (6)

Or, for the linearized case,

$\begin{matrix}{\begin{bmatrix}c_{11} & c_{12} & c_{13} \\c_{21} & c_{22} & c_{23} \\c_{31} & c_{32} & c_{33}\end{bmatrix}_{n + 1} = {\begin{bmatrix}c_{11} & c_{12} & c_{13} \\c_{21} & c_{22} & c_{23} \\c_{31} & c_{32} & c_{33}\end{bmatrix}_{n}\begin{bmatrix}1 & 0 & \theta_{y} \\0 & 1 & {- \theta_{x}} \\{- \theta_{y}} & \theta_{x} & 1\end{bmatrix}}_{n}} & \left( {7\; a} \right)\end{matrix}$

And so, the coordinate system at any location along the array is givenby,

$\begin{matrix}{{\overset{\overset{\_}{\_}}{C}}_{p} = {{{\overset{\overset{\_}{\_}}{C}}_{0}{\overset{\overset{\_}{\_}}{R}}_{0}{\overset{\overset{\_}{\_}}{R}}_{1}{\overset{\overset{\_}{\_}}{R}}_{2}\mspace{14mu} \ldots \mspace{14mu} {\overset{\overset{\_}{\_}}{R}}_{p}} = {{\overset{\overset{\_}{\_}}{C}}_{0}{\prod\limits_{n = 0}^{p}\; {\overset{\overset{\_}{\_}}{R}}_{n}}}}} & (8)\end{matrix}$

The initial value of this coordinate system matrix,

$\begin{matrix}{{\overset{\overset{\_}{\_}}{C}}_{0} = \begin{bmatrix}c_{11} & c_{12} & c_{13} \\c_{21} & c_{22} & c_{23} \\c_{31} & c_{32} & c_{33}\end{bmatrix}_{0}} & (9)\end{matrix}$

describes the initial orientation of the fiber in the exteriorcoordinate system. If the fiber is initially aligned along the z-axis,the matrix will be,

$\begin{matrix}{{\overset{\overset{\_}{\_}}{C}}_{0} = \begin{bmatrix}{\sin \; \beta} & {{- \cos}\; \beta} & 0 \\{\cos \; \beta} & {\sin \; \beta} & 0 \\0 & 0 & 1\end{bmatrix}_{0}} & (10)\end{matrix}$

In the description above, the first two vectors still have one degree offreedom, which is the rotation of the fiber around its axis—the samerotational degree of freedom in the m-matrix above. This is because withthree cores we cannot sense the fiber rotation around its axis. In manyimplementations, this situation is not generally a problem, because itwill generally be taken care of automatically by the way the fiber isembedded in or associated with the kinematic chain and by calibration.Further, it means that complete generality can be retained even if theinitial matrix is restricted to be,

$\begin{matrix}{{\overset{\overset{\_}{\_}}{C}}_{0} = \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}_{0}} & (11)\end{matrix}$

B. Computation Example for Four Cores

For the case of four cores, as illustrated in FIG. 4B for instance, asimilar derivation is possible, and it can be shown that in such a casealso the fiber rotation around its axis can be measured. With fourcores,

Δε_(p,n)=ε_(2,n)−ε_(1,n)   (2a)

Δε_(q,n)=ε_(3,n)−ε_(1,n)   (2b)

Δε_(r,n)=ε_(4,n)−ε_(1,n)   (2c)

where Δε_(p,n), Δε_(q,n), and Δε_(r,n) designate the three differentialstrain arrays. These differential strains are then converted into localbends in an ortho-normal coordinate system using a simple lineartransformation,

$\begin{matrix}{\begin{bmatrix}\theta_{x,n} \\\theta_{y,n} \\\theta_{z,n}\end{bmatrix} = {\begin{bmatrix}m_{px} & m_{qx} & m_{rx} \\m_{py} & m_{qy} & m_{ry} \\m_{pz} & m_{qz} & m_{rz}\end{bmatrix}\begin{bmatrix}{\Delta \; ɛ_{p,n}} \\{\Delta \; ɛ_{q,n}} \\{\Delta \; ɛ_{r,n}}\end{bmatrix}}} & \left( {3b} \right)\end{matrix}$

The m-matrix m is a full description of the multi-core fiber, capturingthe effects of the locations of the cores and the initial rotationalorientation of the fiber in the coordinate system.

These three rotation values are used to create a rotation matrix equalto the product of a first rotation of an angle θ_(x,n) around thex-axis, a second rotation of θ_(y,n) around the y-axis, and a thirdrotation of θ_(z,n) around the z-axis according to the equations:

$\begin{matrix}\begin{matrix}{{\overset{\overset{\_}{\_}}{R}}_{x,n} = \begin{bmatrix}1 & 0 & 0 \\0 & {\cos \; \theta} & {{- \sin}\; \theta} \\0 & {\sin \; \theta} & {\cos \; \theta}\end{bmatrix}} \\{{\overset{\overset{\_}{\_}}{R}}_{y,n} = \begin{bmatrix}{\cos \; {\theta`}} & 0 & {\sin \; \theta} \\0 & 1 & 0 \\{{- \sin}\; \theta} & 0 & {\cos \; \theta}\end{bmatrix}} \\{{\overset{\overset{\_}{\_}}{R}}_{zn} = \begin{bmatrix}{\cos \; \theta} & {{- \sin}\; \theta} & 0 \\{\sin \; \theta} & {\cos \; \theta} & 0 \\0 & 0 & 1\end{bmatrix}} \\{{\overset{\overset{\_}{\_}}{R}}_{n} = {{\overset{\overset{\_}{\_}}{R}}_{x,n}{\overset{\overset{\_}{\_}}{R}}_{y,n}{\overset{\overset{\_}{\_}}{R}}_{z,n}}}\end{matrix} & \left( {4b} \right)\end{matrix}$

For small angle approximation, the above expression simplifies to:

$\begin{matrix}{{\overset{\overset{\_}{\_}}{R}}_{n} = \begin{bmatrix}1 & {- \theta_{z,n}} & \theta_{x,n} \\\theta_{z,n} & 1 & \theta_{y,n} \\{- \theta_{x,n}} & {- \theta_{y,n}} & 1\end{bmatrix}} & \left( {5b} \right)\end{matrix}$

where, because a first order small angle approximation is used, R_(n) isa valid rotation matrix only if θ_(x)<<1 and θ_(y)<<1 and θ_(z)<<1.

If sufficiently small spatial increments are used, the above conditionsare not difficult to satisfy. This rotation matrix is then moved intothe coordinate system at the n^(th) position on the fiber. In this way,the calculations are iterated to walk down the length of the fiber,reconstructing the tangent vector, as well as the vectors defining therotational coordinate system, along the way. The iterative equation isthe same as for the three core case,

C _(n+1) = C _(n) R _(n)   (6)

Or, for the linearized case,

$\begin{matrix}{\begin{bmatrix}c_{11} & c_{12} & c_{13} \\c_{21} & c_{22} & c_{23} \\c_{31} & c_{32} & c_{33}\end{bmatrix}_{n + 1} = {\begin{bmatrix}c_{11} & c_{12} & c_{13} \\c_{21} & c_{22} & c_{23} \\c_{31} & c_{32} & c_{33}\end{bmatrix}_{n}\begin{bmatrix}1 & {- \theta_{z}} & \theta_{y} \\\theta_{z} & 1 & {- \theta_{x}} \\{- \theta_{y}} & \theta_{x} & 1\end{bmatrix}}_{n}} & \left( {7b} \right)\end{matrix}$

And so, the coordinate system at any location along the array is givenby,

$\begin{matrix}{{\overset{\overset{\_}{\_}}{C}}_{p} = {{{\overset{\overset{\_}{\_}}{C}}_{0}{\overset{\overset{\_}{\_}}{R}}_{0}{\overset{\overset{\_}{\_}}{R}}_{1}{\overset{\overset{\_}{\_}}{R}}_{2}\ldots {\overset{\overset{\_}{\_}}{R}}_{p}} = {{\overset{\overset{\_}{\_}}{C}}_{0}{\prod\limits_{n = 0}^{p}{\overset{\overset{\_}{\_}}{R}}_{n}}}}} & (8)\end{matrix}$

The initial value of this coordinate system matrix,

$\begin{matrix}{{\overset{\overset{\_}{\_}}{C}}_{0} = \begin{bmatrix}c_{11} & c_{12} & c_{13} \\c_{21} & c_{22} & c_{23} \\c_{31} & c_{32} & c_{33}\end{bmatrix}_{0}} & (9)\end{matrix}$

describes the initial orientation of the fiber in the exteriorcoordinate system. If the fiber is initially aligned along the z-axis,complete generality can be retained even if the initial matrix isrestricted to be,

$\begin{matrix}{{\overset{\overset{\_}{\_}}{C}}_{0} = \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}_{0}} & (11)\end{matrix}$

C. Computation Example for Three and Four Cores

The mathematical treatment that proceeds from now on is identical forboth the three and four core fiber cases in the above examples.

The tangent vector {right arrow over (t)} is the last column of the Cmatrix,

$\begin{matrix}{\overset{\_}{t} = {\overset{\overset{\_}{\_}}{C} \cdot \begin{bmatrix}0 \\0 \\1\end{bmatrix}}} & (12)\end{matrix}$

Accordingly, the tangent vector at any particular point is the productof all of the previous rotation vectors,

$\begin{matrix}{{\overset{\_}{t}}_{p} = {{\overset{\overset{\_}{\_}}{C}}_{0}{\prod\limits_{n = 0}^{p}{{\overset{\overset{\_}{\_}}{R}}_{n} \cdot \begin{bmatrix}0 \\0 \\1\end{bmatrix}}}}} & (13)\end{matrix}$

The position at any point along the fiber is the sum of all of theprevious tangent vectors, multiplied by the length of fiber that theyrepresent,

$\begin{matrix}{\begin{bmatrix}x \\y \\z\end{bmatrix}_{q} = {\Delta \; d{\sum\limits_{p = 0}^{q}{\overset{->}{t}}_{p}}}} & (14)\end{matrix}$

Substituting in the expression for the tangent vector gives,

$\begin{matrix}{\begin{bmatrix}x \\y \\z\end{bmatrix}_{q} = {\Delta \; d{\sum\limits_{p = 0}^{q}\left\lbrack {\left\{ {{\overset{\overset{\_}{\_}}{C}}_{0}{\prod\limits_{n = 0}^{p}{\overset{\overset{\_}{\_}}{R}}_{n}}} \right\} \cdot \begin{bmatrix}0 \\0 \\1\end{bmatrix}} \right\rbrack}}} & (15)\end{matrix}$

For generality, an arbitrary offset vector can be added to place thecalculated coordinates into any arbitrary coordinate system.

$\begin{matrix}{{\begin{bmatrix}x \\y \\z\end{bmatrix} = {{\Delta \; d{\sum\limits_{p = 0}^{q}\left\lbrack {\left\{ {{\overset{\overset{\_}{\_}}{C}}_{0}{\prod\limits_{n = 0}^{p}{\overset{\overset{\_}{\_}}{R}}_{n}}} \right\} \cdot \hat{z}} \right\rbrack}} + {{\overset{\_}{v}}_{0}\mspace{14mu} {where}}}},} & (16) \\{{{\overset{\overset{\_}{\_}}{v}}_{0} = {\begin{bmatrix}x_{0} \\y_{0} \\z_{0}\end{bmatrix}\mspace{14mu} {and}}},} & (17) \\{\hat{z} = \begin{bmatrix}0 \\0 \\1\end{bmatrix}} & (18)\end{matrix}$

For the computation of the position and orientation of the frame ofreference at the end of a segment with respect to the frame of referenceat the start of the segment, C ₀ is the identity matrix, and {rightarrow over (ν)}₀ is a vector of zeros, which represents the frame ofreference at the start of the segment. Alternatively, the computationcan be carried in another base or world frame located, for instance, atthe base of the kinematic chain. In this case C ₀ is the 3×3 matrixspecifying the orientation of the frame of reference at the start of thesegment with respect to the above-mentioned base frame, and {right arrowover (v)}₀ is the 3×1 vector specifying the position of the origin ofthe frame of reference at the start segment with respect to theabove-mentioned base frame.

As mentioned above, in some instances the quantity Δd is known from theproperty of the particular interferometer that is used. Alternatively,Δd can be calibrated by laying the segment of fiber in a straight line,for instance with the use of a fixture, and comparing the computedsegment tip position from equation 18 with the known segment physicallength.

II. Illustrative Implementations and Embodiments

FIGS. 7A, 7B, and 7C are diagrammatic views of an illustrative kinematicchain 700 that includes two elongate, rigid mechanical links 702 a, 702b interconnected by one or more mechanical constraints. The one or moremechanical constraints are illustrated as a flexible mechanicalstructure 704, for example. In one illustrative embodiment, flexiblestructure 704 is a wrist mechanism, as described in U.S. Pat. No.6,817,974 (filed 28 Jun. 2002), which is incorporated herein byreference. Briefly, servomotor actuators control distally anchoredcables that run through a flexible wrist assembly so as to move thewrist in one or two DOF's (e.g., pitch; pitch and yaw). The controlcables are omitted from FIGS. 7A-7C for clarity.

As shown in FIG. 7A, a three-core, continuously EBG-configured opticalfiber 706 as described above is routed through proximal mechanical link702 a, flexible structure 704, and distal mechanical link 702 b. Fiber706 is shown as being aligned with the longitudinal centerline of links702 and flexible structure 704. In one embodiment, fiber 706 ispositioned inside a flexible sheath (e.g., Teflon® FEP) (not shown) thatis positioned within central lumens of 702 a, 702 b, and 704. Due to itscenterline alignment, fiber 706 remains in position and does not slidelongitudinally (surge) with reference to the links 702 when flexiblestructure 704 bends, as shown in FIG. 7B. Accordingly, fiber 706 may ormay not be fixed in position (e.g., glued or otherwise affixed) relativeto either or both links 702 a, 702 b.

In some embodiments a resilient spring 707 is routed through themechanical structure 704 to keep the small links in a minimum energyconfiguration, as described in more detail below. The spring 707 may be,for example, a helical wound coil or a resiliently bendable tube (e.g.,of superelastic material), and in some instances such springimplementations may act as a conduit through mechanical structure 704for the fiber. Skilled individuals will understand how the spring actson the small links as mechanical structure 704 bends, although forclarity spring 707 is not shown in FIGS. 7B and 7C. Additional detailsof mounting and routing a shape-sensing optical fiber with reference toa link are described in concurrently filed U.S. Pat. Appl. Ser. No.12/164,297 (filed 30 Jun. 2008), which is incorporated herein byreference.

To account for small mechanical variations in kinematic chain 700 andfor inaccuracies during assembly, the segment 708 that is used for shapesensing is defined to have a segment start L_(1S) within proximal link702 a (i.e., proximal to the proximal end of flexible structure 704).Similarly, segment 708's segment end L_(1E) is defined within distallink 702 b (i.e., distal to the distal end of flexible structure 704).Defining the segment start L_(1S) and segment end L_(1E) within therigid links ensures that the position and orientation of segment endL_(1E) encompasses the complete bend for flexible structure 704.Further, due to real world mechanics, if fiber 706 strays from itscenterline alignment as structure 704 bends, perhaps causing fiber 706to slide inside either or both segments 702 a, 702 b, then the positionand orientation of segment end L_(1E) will still provide validinformation for the pose of kinematic structure 700. For example, ifsegment 708 of fiber 706 follows an alternate path 714 as shown in FIG.7B, the position and orientation information at segment end L_(1E) isvalid.

In some instances, however, the fiber may be offset from the kinematicchain's longitudinal centerline. As shown in FIG. 7A, for example, anillustrative fiber 720 is routed through links 702 a, 702 b and flexiblestructure 704 away from the centerlines. Fiber 720's shape sensingsegment 722 is defined between segment start L_(2S) within proximal link702 a and segment end L_(2E) within distal link 702 b. For sufficientcenterline offsets and bend angles, fiber 720 will slide as flexiblestructure 704 bends. For example, if fiber 720 is fixed in relation toproximal link 702 a, then segment end L_(2E) translates distally insidedistal link 702 b as flexible structure 704 bends, as shown in FIG. 7B.Similarly, segment end L_(2E) translates proximally inside distal link702 b as flexible structure 704 bends in an opposite direction, as shownin FIG. 7C. Similar sliding of segment start L_(2S) may occur if thefiber is fixed in the distal link 702 b, or if the fiber is free toslide through both rigid links. As long as the segment start L_(2S) andsegment end L_(2S) remain within their links, valid position andorientation information can be determined.

FIG. 8 is a schematic view of a minimally invasive surgical instrument800 that is representative of various such instruments used in the daVinci® Surgical System, manufactured by Intuitive Surgical, Inc. ofSunnyvale, Calif., and modified in accordance with aspects of theinvention. Instrument 800 includes an elongate, hollow shaft 802, awrist mechanism 804 coupled to the distal end of shaft 802, a smalldistal link 806 coupled to the distal end of wrist mechanism 804, and asurgical end effector 808 (e.g., grasper, needle driver, shears, cauterytool, camera, and the like) coupled to the distal end of link 806. Aforce transmission mechanism 810 is coupled to the proximal end of shaft802. Teleoperated servomotors engage transmission mechanism 810components (e.g., rotating disks), which in turn pass activating forcesthrough shaft 802 via cables and/or cable/hypotube assemblies to movewrist mechanism 804 and end effector 808. Additional illustrativedetails may be found, e.g., in U.S. Pat. No. 6,817,974 referenced above,and in U.S. Pat. No. 5,807,377 (filed 16 May 1997) and U.S. Pat. No.6,461,372 (filed 8 Mar. 2000), which are incorporated herein byreference. In accordance with aspects of the invention, instrument 800includes a three-core optical fiber 812 for shape sensing as describedabove.

The proximal end of fiber 812 is affixed in an illustrative connector814, to be attached to a fiber strain interrogator unit. Connector 814is further illustrative of embodiments in which each core of fiber 812is coupled to an individual single core fiber, and each individual fiberis coupled to the interrogator unit. See e.g., U.S. Pat. App. Publ. No.US 2006/0013523 A1 referenced above.

Fiber 812 is then routed through, e.g., transmission mechanism 810,shaft 802, wrist mechanism 804, so that fiber 812's distal endterminates, e.g., at or near the distal end of link 806. The fiber mayenter the instrument at various other positions, e.g., in thetransmission mechanism or the shaft. As described above, in one aspect abase frame is defined at the proximal end 818 of fiber 812. A segmentstart L_(1S) is defined in fiber 812 at a location proximal of wristmechanism 804, and a segment end L_(1E) is defined in fiber 812 at alocation distal of wrist mechanism 804 (e.g., at distal end 824 of fiber812. As wrist mechanism 804 bends, the position and orientation ofsegment end L_(1E) is determined with reference to segment start L_(1S).Therefore, the position and orientation of link 806 is determined withreference to shaft 802.

FIG. 9 is a schematic view of another illustrative minimally invasivesurgical instrument 900. Surgical instrument 900 includes severalcomponents (e.g., shaft, cables, wrist mechanism, end effector) similarto those of surgical instrument 800. Surgical instrument 900 alsoincludes a parallel motion mechanism 902 between shaft 904 and link 906,in addition to a flexible wrist mechanism 908 between link 906 anddistal end link 910. Parallel motion mechanism 902 functions to changelink 906's position (e.g., heave, sway) without changing link 906'sorientation with respect to shaft 904. FIG. 9A is an illustrativeschematic of parallel motion mechanism 902 in a displaced pose, andFIGS. 9B and 9C are elevation views of an illustrative embodimentshowing the parallel motion mechanism in line with shaft 904 (FIG. 9B)and displaced in position (FIG. 9C). Details of parallel motionmechanism 902 are described in U.S. Pat. Appl. Pub. No. US 2008/0065102A1 (filed 13 Jun. 2007), which is incorporated herein by reference.

Referring to FIG. 9, multi-core fiber 912 is routed through theinstrument 900 components in a manner similar to that described forinstrument 800. In one aspect, a single shape sensing segment is definedbetween segment start L_(1S) and segment end L_(1EA). This single shapesensing segment includes parallel mechanism 902, link 906, wristmechanism 908, and at least a portion of distal link 910. Alternatively,a first shape sensing segment is defined between segment start L_(1S)and segment end L_(1EB), and a second shape sensing segment is definedbetween segment start L_(2S) and segment end L_(2E). The first shapesensing segment is used to determine changes in position and orientationthat result from moving parallel motion mechanism 902. The second shapesensing segment is used to determine changes in position and orientationthat result from moving wrist mechanism 908. As shown in FIG. 9, in oneaspect segment end L_(1EB) and segment start L_(2S) are not coincident.In this aspect the spatial relationship between segment end L_(1EB) andsegment start L_(2S) is known because they are both in an unchangingrelationship within rigid link 906. In some aspects, however, a segmentend and a segment start may be coincident or may overlap, as describedabove.

For a kinematic chain with long, rigid links, data processing resources,and more importantly the total length of FBG's in the fiber, may beconserved by defining shape sensing segments to cover only the chain'sbendable parts, as illustrated in FIGS. 3B and 3C, or as in FIGS. 3D and3E. There is no need to process strain information to determine theunchanging shape of such links, and a priori mechanical information inconjunction with the information from the shape sensing segments can beused to determine the chain's pose.

Since the position and orientation of the distal end of instrument 900is determined by sensing the shape of a fiber segment, since themechanical characteristics (e.g., link dimensions, positions andorientations of rotational axes in revolute joints, and the like) of thekinematic chain in the instrument are known, and since the kinematicchain does not contain redundant degrees of freedom, inverse kinematiccalculations are used to determine the instrument's pose.

FIG. 10 is a diagrammatic view of a kinematic chain 1000 that includestwo links 1002 a, 1002 b interconnected by a revolute joint 1004. Ashape sensing optical fiber 1006 in accordance with aspects of theinvention is shown in phantom line passing through link 1002 a, joint1004, and link 1002 b. As shown in FIG. 10, a portion 1008 of fiber 1006moves freely within joint 1004. Shape sensing as described herein isused to determine position and orientation of segment end L_(E) definedwithin link 1002 b with reference to segment start L_(S) defined withinlink 1002 a. This embodiment is illustrative of shape sensing asdescribed herein applied to various kinematic chain configurations. Itcan be seen that a similar configuration can be used for a prismaticjoint between two links, as long as a sufficient loop of fiber isavailable to accommodate the prismatic joint's movement. And, aspects ofthe invention are not limited to kinematic chains that have long, rigidlinks coupled by single-axis revolute joints.

Another class of medical devices, such as endoscopes or catheter guidedevices, are based on flexible structures, as described above.Snake-like structures with short, discrete components can be effectivelyapproximated as a continuously bending element of constant or variablelength as determined by the loads on the structure's longitudinal(end-to-end) axis.

In such snake-like mechanisms, the multi-core fiber can be ran along orinside the device's centerline in a way similar to that described abovewith reference to FIG. 7A, and consequently the fiber does not need toslide as the structure bends. Alternatively, the fiber can run (e.g., ina conduit) at an offset from and parallel to the centerline in a mannersimilar to that described above with reference to FIGS. 7B and 7C, or itmay run along a helicoidal path around the centerline.

For snake-like mechanisms, the number of controlled DOF's is usuallymuch smaller than the number of actual mechanical links, and thus themechanism's actual DOF's are redundant. In some cases the redundancyproblem is solved by using a central spring element that forces thelinks to assume the minimum energy configuration compatible with thecontrolled DOF's (at least absent external forces along the snake bodyand statically). For example, a pair of controlled DOF's under thisconfiguration can set the final orientation change over a set ofcontiguous links in the pitch and yaw directions. Therefore, forflexible devices (both snake-like and continuously bending), a shapesensing segment is defined for each length of the device that iscontrolled by one or more DOF's. For example, if a 10 cm length of thedevice is controllable in pitch, and an immediately adjacent 10 cmlength of the device is controllable in yaw (or pitch, or some otherorientation or position), then each controllable length of the devicewill have a corresponding shape sensing segment defined. If a 10 cmlength is controllable in both pitch and yaw, then the length will stillhave a single corresponding shape sensing defined. If controllablelengths of a device overlap (e.g., a length controllable in yaw beginsat a mid-point of a length controllable in pitch), then each overlappingcontrollable length will have a corresponding shape sensing segmentdefined.

FIG. 11 is a schematic view of a kinematic chain 1100 that isillustrative of kinematic chains made of one, two, or more flexiblemechanical sections. As shown in FIG. 11, kinematic chain 1100 has abase section 1102, four flexible sections 1104 a-1104 d coupled in linedistally of base section 1102, and an illustrative distal end effector1106 (e.g., grasper, needle driver, shears, cautery tool, camera, andthe like) coupled to the distal end of the most distal section 1104 d.In some cases, the end effector may be omitted, and kinematic chain 1100may function as a guide that allows other mechanical devices to berouted through, over, or adjacent chain 1100. An actuator mechanism 1108is coupled to the proximal end of base section 1102. Actuator mechanism1108 includes components (e.g., servomotors, hand-operated wheels)coupled to cables (not shown) that are coupled to the various flexiblesections 1104. Tension on the cables causes associated flexible sections1104 to bend in one or more directions. U.S. Pat. No. 4,873,965 (filed15 Jul. 1988) and U.S. Pat. No. 5,174,276 (filed 28 Jun. 1991), whichare incorporated herein by reference, are illustrative of suchmechanisms.

In a manner similar to that described above, a three-core shape sensingoptical fiber 1110 is routed through the base section 1102 and theflexible sections 1104 a-1104 d. A proximal end connector 1112 is usedto connect fiber 1110 to an interrogator unit, as described above. Inaccordance with aspects of the invention, at least four shape sensingsegments are defined in fiber 1110. Each of the shape sensing segmentsis associated with one of the flexible sections 1104. Shape sensingsegment 1114 a is associated with flexible section 1104 a and starts atsegment start L₀, thereby demonstrating that a shape sensing segment maybegin at the origin of the base reference frame. Likewise, segment 1114b is associated with flexible section 1104 b and starts at segment startL₁, segment 1114 c is associated with flexible section 1104 c and startsat segment start L₂, and segment 1114 d is associated with flexiblesection 1104 d and starts at segment start L₃. As described above,several of the segment starts and ends may be defined at the samelocation in fiber 1110, or short, additional segments may be definedbetween the various segment ends and starts.

As described above, the position and orientation of each segment end isdetermined with reference to a reference frame at its correspondingsegment start. This position and orientation information is aggregatedto determine the position and orientation of the distal end of section1104 d (and therefore, in this example, end effector 1106) withreference to a base reference frame associated with base 1102. Further,since the position and orientation of the distal end of the flexibledevice in FIG. 11 is determined by sensing the shape of a fiber segment,since the mechanical characteristics (e.g., link dimensions, positionsand orientations of rotational axes in revolute joints, and the like) ofthe kinematic chain in the snake device are known, and since thekinematic chain redundant DOF's can be computed to minimize the centerspine elastic potential energy, inverse kinematic calculations are usedto determine the flexible device's pose. Determining the pose isimportant, for example, in surgical applications in which it isnecessary to avoid collision with another instrument or with certaintissue structures.

In all the cases described above and depicted in FIGS. 7-11, presentedin accordance with aspects of the invention, the position of thekinematic chain is estimated by combining three sources of information;(i) the Cartesian information produced by the shape sensor for each ofthe defined segments; (ii) the a priori knowledge of the kinematic modelof the kinematic chain (e.g., as stored in a readable electronicmemory); and (iii) the a priori knowledge of the nature of themechanical constraints between the kinematic chain and the shape sensingfiber at the start and end of the segment (e.g., as stored in a readableelectronic memory). A general description of the methodology ispresented and several relevant applications are described below.

The Cartesian information produced by the shape sensor consists of theposition and orientation of the end point of each of the definedsegments with respect to each segment's initial frame of reference. Theposition is a 3D vector {right arrow over (p)}_(s), and the orientationis represented for instance by the tangent vector {right arrow over(t)}_(s). An additional roll angle r_(s) can be used to specify therotation of the fiber around its axis at the end of the segment. Allthree quantities are computed for each segment as described in equations13 and 15 above.

The a priori knowledge of the kinematic model of the kinematic chain isrepresented by a forward kinematic model that allows computing theCartesian position and orientation of the frame of reference at the endof the segment as a function of a vector {right arrow over (q)} of njoint variables [{right arrow over (q)}₁ . . . {right arrow over(q)}_(n)] according to:

[p _(k) t _(k)]=ƒkin({right arrow over (q)})   (19)

The kinematic model of a kinematic chain is easily constructed accordingto known methods. For instance, the procedure described in John J.Craig, Introduction to Robotics: Mechanics and Control (PearsonEducation Ltd. 2004), which is incorporated herein by reference, may beused. Denavit Hartenberg frames of reference are assigned to each jointof the chain, the base frame of reference is defined as the segmentstart frame of reference, and the tool tip frame of reference is definedas the segment end frame of reference. Both rotational and translational(prismatic) joints can be easily handled by this well-known method. Thenumber of joints spanned by the defined segment is arbitrary. Asexamples, the following cases are described below: (i) a single DOF, asin the joint illustrated by FIG. 10; (ii) two DOF's, as in the wristmechanisms illustrated in FIGS. 7A, 7B, 7C, and 8; (iii) two DOF's, asin the mechanism illustrated in FIG. 9A; (iv) four DOF's, as in themechanisms illustrated in FIGS. 9-9C; (v) six DOE's; and (vi) large (>6)numbers of DOF's as in the mechanism illustrated in FIG. 11. In thislast case, the kinematic model is completed by a scalar function thatrepresents the potential energy of the spring backbone as a function ofthe joint variables {right arrow over (q)}: E({right arrow over (q)}).

The a priori knowledge of the nature of the mechanical constraintsbetween the kinematic chain and the fiber at the start and the end ofthe segment enables the following fiber attachment cases to bedistinguished. In the first case, the fiber is mechanically attached atboth ends of the segment. In this first case, the fiber shape sensor ismeasuring a position and orientation of the end of the fiber segmentthat is directly related to the position and orientation of the link ofthe kinematic chain embedding the end of the fiber segment. Hence,information about the six DOF's of the links is obtained if a twistsensing fiber is used, while information about five DOF's of the link isobtained if a 3 core fiber is used. In the second case, the fiber isattached at the start of the segment and free to slide at the end of thesegment, for example in a cylindrical conduit embedded in a link of thekinematic chain. In this second case the fiber shape sensor is measuringthe position and orientation in space of the conduit in the link of thekinematic chain embedding the end of the fiber segment. If the conduitis cylindrical and tightly fits around the fiber, then information aboutfour DOF's of the links is obtained. The third case is similar to thesecond case, except the fiber is attached at the end of the segment andis free to slide at the start of the segment. As in the second case,information about four DOF's of the links is obtained.

In the three illustrative fiber attachment cases above, the fiber is notrequired to go through the centerline of the kinematic chain or along apath specified in any way. If assumptions can be made about the pathfollowed by the fiber, such as that the path does not change in lengthor that the path lies in a plane or in a set of planes, then morespecialized approaches described below can be also used. For example, ifthe start or end of the fiber is guided within the links of thekinematic chain in slits that tightly constrain the fiber in only aplane of the embedding links, then information about three DOF's of thelinks is obtained.

The three sources of information described above are combined togetherin a goal function that is then numerically minimized to compute thebest estimate for the joint variables {right arrow over (q)}.

The minimization of the nonlinear goal function can be achieved withknown techniques, such as gradient descent methods, Newton Rapsonmethods, and other methods as described, for instance, in Deepak Tolani,Ambarish Goswami, & Norman I. Badler, Real-Time Inverse KinematicsTechniques for Anthropomorphic Limbs, 62 Graphical Models 353-388(September 2000) and in N. I. Badler, K. Manoochehri, & G. Walters,Articulated Figure Positioning by Multiple Constraints, 7 IEEE ComputerGraphics and Applications 28-38 (June 1987), which are incorporatedherein by reference. The minimization results are different depending onthe relationship between the number of DOF's of the kinematic chain andthe number of DOF's of information that the shape sensor is able togather about the link.

In particular, three sensed/actual DOF cases can be distinguished. Inthe first sensed/actual DOF case, the number of kinematic DOF's is lowerthan the number of sensed DOF's. In this first case the position {rightarrow over (q)} of the kinematic chain can be determined and also thenoise in the {right arrow over (q)} from the sensor can be minimized bymerging together the redundant sensor information, which can also beweighed differently according to its noise to signal ratio. In thesecond sensed/actual DOF case, the number of kinematic DOF's is equal tothe number of sensed DOF's. In this second case the minimization isconverging to a unique solution for the vector {right arrow over (q)},and the minimization algorithm is a kind of inverse kinematics algorithmthat is well described in the robotics literature (see e.g., SpringerHandbook of Robotics (Bruno Siciliano & Oussama Khatib eds., Springer2008), which is incorporated herein by reference). In the thirdsensed/actual DOF case, the number of kinematic DOF's is higher than thenumber of sensed DOF's. In such a case, the vector {right arrow over(q)} cannot be uniquely determined (strictly speaking). In this thirdcase, therefore, the kinematic chain should be split into two or moreparts, and for each part a shape sensing segment of fiber should bedefined to produce position and orientation information so that noambiguity is possible about the joint positions.

Further, in this third sensed/actual DOF case, two special subcases canbe solved by using just one sensing segment of fiber. The first subcaseconcerns a kinematic chain representing flexible kinematics of a devicewith a central spring. In this first subcase, the potential energyE({right arrow over (q)}) of the spring can be minimized as part of theoverall minimization, and thus the whole vector {right arrow over (q)}can be uniquely determined under the restrictive hypothesis of noexternal forces acting on the mechanical structure. The second subcaseconcerns a kinematic chain whose links have constraints to their motionsthat are in addition to the ones from the joints. For example, there maybe constraints due to mechanical construction, such as from cablesrunning through the links with coupled motions, as in a snake-like wristcomposed of four links and four joints but having only two degrees offreedom as a result of the kinematic coupling enforced by the actuationcables. In this second subcase, the additional constraints reduce thekinematic DOF's, and the case is reduced to the first or secondsensed/actual DOF cases described above.

The following specific examples illustrate the outlined methods forselected cases. It should be understood that these examples are notlimiting, and that aspects of the invention may be applied in other thanthe selected cases below.

A. EXAMPLE 1

Referring to FIG. 7, a kinematic chain of four links coupled to retaintwo DOF's of bending can be described by an overall kinematic model:

[p _(k) t _(k)]=ƒkin({right arrow over (q)} ₁ ,{right arrow over (q)} ₂)  (20)

The sensing segment start frame is defined in the first link, and thesensing segment end frame is defined in the second link. The three casesof the sensing fiber attached to both links, attached in the first linkand unconstrained at the tip to slide in a tight cylindrical conduit inthe second link, or attached in the second link and unconstrained toslide in a tight cylindrical conduit in the first link, can all bedescribed by the goal function:

Min with respect to {right arrow over (q)} ₁ ,{right arrow over (q)} ₂of {Abs(t _(s) −t _(k))}  (21)

This function describes that the fiber segment end is constrained tohave the same orientation with respect to the fiber segment start as thesecond link has with respect to the first link. Given that theorientation measurement t_(s) has two DOF's (it is a unit vector),{right arrow over (q)}₁ and {right arrow over (q)}₂ are uniquelydetermined.

B. EXAMPLE 2

Referring to FIG. 10, a single-DOF rotational joint {right arrow over(q)}₁ can be described by the kinematic model of the form:

[p _(k) t _(k)]=ƒkin({right arrow over (q)} ₁)   (22)

The sensing segment start frame is defined in the first (proximal) link,and the sensing segment end frame is defined in the second (distal)link. The sensing fiber may be attached in the first link andunconstrained to slide in a tight, cylindrical conduit in the secondlink, or the sensing fiber may be attached in the second link andunconstrained to slide in a tight, cylindrical conduit in the firstlink. The associated goal function is:

Min with respect to {right arrow over (q)}₁ of {Abs(t _(s) −t_(k))}  (23)

Given that the orientation measurement t_(s) has two DOF's (it is a unitvector), {right arrow over (q)}₁ is determined, and the measurementnoise can be minimized too.

In another implementation, the fiber may be constrained in a slit(rather than in a cylindrical conduit) in either or both of the proximaland distal links, with the slit(s) oriented parallel to the joint's axisof rotation. Thus, the fiber may slide sideways in the slit as well asalong the link's longitudinal axis. The associated goal function is:

Min with respect to {right arrow over (q)} ₁ of {Abs└(t _(s) −t _(k))−(t_(s) −t _(k))·z_(j)┘}  (24)

where · indicates the scalar product. In this case, {right arrow over(q)}₁ is uniquely determined.

C. EXAMPLE 3

Referring to FIG. 9A, mechanism 902 illustrates a kinematic chain ofseven links and additional mechanical constraints that enforce theparallelism of the output link 906 with respect to shaft 904 so that twoDOF's {right arrow over (q)}₁ and {right arrow over (q)}₂ are sufficientto describe the overall kinematics according to a model of the form:

[p _(k) t _(k)]=ƒkin({right arrow over (q)} ₁ ,{right arrow over (q)} ₂)  (25)

The sensing segment start frame is defined in the first link (shaft 904)and the sensing segment end frame is defined in the second link (link906). The sensing fiber may be attached in the first link andunconstrained to slide in a tight, cylindrical conduit in the secondlink, or the sensing fiber may be attached in the second link andunconstrained to slide in a tight, cylindrical conduit in the firstlink. The associated goal function that describes that the fiber endsegment position with respect to the fiber start segment position in theplane orthogonal to the fiber axis is equal to the position of secondlink with respect to the first link in the plane orthogonal to the fiberaxis is:

Min with respect to {right arrow over (q)} ₁ ,{right arrow over (q)} ₂of {Abs[(p _(s) −p _(k))−(p _(s) −p _(k))·z ₁]}  (26)

with z₁ being the unit vector of the z-axis of the segment start frame.In such a case, {right arrow over (q)}₁ and {right arrow over (q)}₂ aredetermined uniquely from the two DOF's of measurement.

Referring now to the case illustrated in combined FIGS. 9-9C, therequired position and orientation information can be sensed by definingtwo fiber sensing segments. The first segment senses the configurationof parallel motion mechanism 902, as described above. The second segmentsenses the configuration of wrist mechanism 908 (between links 906 and910) with the methodology already described for FIG. 7.

Alternatively, especially if the two mechanisms blend into one anotherover a short distance, the configuration of the entire distal end of theinstrument can be sensed by using a single sensing segment. Thekinematic chain describing the whole distal end has two DOF's inmechanism 902 and two more in mechanism 908, as described above.Consequently, the kinematic model is specified by four kinematicparameters {right arrow over (q)}₁,{right arrow over (q)}₂,{right arrowover (q)}₃,{right arrow over (q)}₄ in the form:

[p _(k) t _(k)]=ƒkin({right arrow over (q)} ₁ ,{right arrow over (q)} ₂,{right arrow over (q)} ₃ ,{right arrow over (q)} ₄)   (27)

The sensing segment start frame is defined in shaft 902, and the sensingsegment end frame is defined in distal link 910.

In this single-segment, four-DOF aspect, there are various ways ofrouting and constraining the fiber through the mechanisms, andaccordingly a unique approach to the calculations is developed for eachof the various ways of routing and constraining the fiber. In a firstillustrative approach, the fiber is attached to both shaft 902 and link910 and runs through a center lumen of the instrument. The goal functiondescribing that the relative orientation and position of the fibersegment end with respect to the fiber segment start is identical to therelative orientation and position of the shaft link 902 with respect tothe distal link 910 is:

Min with respect to {right arrow over (q)} ₁ ,{right arrow over (q)} ₂,{right arrow over (q)} ₃ ,{right arrow over (q)} ₄ of {w _(p) ·Abs(p_(s) −p _(k))+w ₁ ·Abs(t _(s) −t _(k))}  (28)

with w_(p) and w_(t) being scalar weights that represent the relativeconfidence in the position data with respect to the orientation data.Accordingly, the tricore shape sensing fiber provides four DOF's ofmeasurement that enable finding the {right arrow over (q)}₁,{right arrowover (q)}₂,{right arrow over (q)}₃ and {right arrow over (q)}₄ valuesand minimizing the noise.

In a second illustrative approach to routing and constraining the fiber,the fiber is allowed to slide in a tight cylindrical conduit in link 910and is not required to run through a center lumen of the instrument. Thegoal function that describes that the information about the position ofthe fiber segment end along the fiber axis is unconstrained from thelink 910 position along the same axis is:

Min with respect to {right arrow over (q)}₁,{right arrow over(q)}₂,{right arrow over (q)}₃,{right arrow over (q)}₄ of

$\begin{matrix}\left\{ {{w_{p} \cdot {{Abs}\left\lbrack {\frac{\left( {p_{s} - p_{k}} \right)}{{Abs}\left( {p_{s} - p_{k}} \right)} - t_{s}} \right\rbrack}} + {w_{l} \cdot {{Abs}\left( {t_{s} - t_{k}} \right)}}} \right\} & (29)\end{matrix}$

In such a case, the fiber provides four DOF's of measurement whichuniquely define the {right arrow over (q)}₁,{right arrow over(q)}₂,{right arrow over (q)}₃, and {right arrow over (q)}₄ values.

In a third illustrative approach to routing and constraining the fiber,the fiber is allowed to slide in a tight cylindrical conduit in shaftlink 902, is attached to link 910, and is not required to run through acenter lumen of the instrument. The goal function that describes thatthe information about the position of the fiber segment start along thefiber axis is unconstrained from the link 902 position along the sameaxis is:

Min with respect to {right arrow over (q)}₁,{right arrow over(q)}₂,{right arrow over (q)}₃,{right arrow over (q)}₄ of

$\begin{matrix}\left\{ {{w_{p} \cdot {{Abs}\left\lbrack {\frac{\left( {p_{s} - p_{k}} \right)}{{Abs}\left( {p_{s} - p_{k}} \right)} - z_{0}} \right\rbrack}} + {w_{t} \cdot {{Abs}\left( {t_{s} - t_{k}} \right)}}} \right\} & (30)\end{matrix}$

where z₀ is the z-axis vector of the frame of reference attached to thefiber segment start. In this approach, the fiber also provides fourDOF's of measurement which uniquely define the {right arrow over(q)}₁,{right arrow over (q)}₂,{right arrow over (q)}₃, and {right arrowover (q)}₄ values.

D. EXAMPLE 4

Referring now to FIG. 11, a kinematic chain with two or more flexiblesections 1104 is shown. Each flexible section has more than six DOF'sand can assume, for example, an S-shape. In each section, some of theDOF's are controlled by an actuator, for instance by means of cablesdriven by a teleoperated servomotor, while the movements in other DOF'sare determined by the interaction of the constraints imposed by thecables and the action of a center spring element. In one implementation,the control cables set the bending of the most distal link of a flexiblesection 1104 (as illustrated in FIG. 11, this most distal link may bethe most proximal link of the next adjacent flexible section), while theorientation of the other intermediate links in the section 1104 are setby the spring force. A kinematic model can be used to describe just theorientation of the most distal link as a function of two joint variables{right arrow over (q)}₁ and {right arrow over (q)}₂ when the springenergy is minimized. This kinematic model takes the form:

[p _(k) t _(k)]=ƒkin({right arrow over (q)} ₁ ,{right arrow over (q)} ₂)  (31)

The sensing fiber may be attached in the first (most proximal) link andunconstrained to slide in a tight, cylindrical conduit in the second(most distal) link, or the sensing fiber may be attached in the secondlink and unconstrained to slide in a tight, cylindrical conduit in thefirst link. The associated goal function that describes that the fibersegment end has the same orientation with respect to the fiber segmentstart that the first link has with respect to the second link is:

Min with respect to {right arrow over (q)}₁,{right arrow over (q)}₂ of{Abs(t_(s)−t_(k))}  (32)

In this way the joint variables {right arrow over (q)}₁ and {right arrowover (q)}₂ can be uniquely determined while the orientation of theintermediate links is assumed to minimize the spring energy.

III. Illustrative System Implementations and Embodiments

FIG. 12 is a diagrammatic view of a shape sensing system. In anillustrative aspect, the proximal end of a shape sensing, multi-core(e.g., 3-core) fiber 1202 as described above is coupled to an OBRinterrogator 1204 using a coupling as described above to establish abase reference frame for fiber 1202. In one implementation, OBR 1204 isan “Optical Backscatter Reflectometer” and associated software purchasedfrom Luna Innovations Incorporated, Roanoke, Va. that is included foreach core of the shape sensing fiber to perform the core strainmeasurements based on the measurement of the phase derivative of thereflected light in each core. Each OBR 1204 provides a core strainmeasurement vector ε(Δdn) to a fiber shape measurement processor 1208that provides additional electronic data processing in accordance withaspects of the invention described in equations 1 to 18 above. In anillustrative embodiment, one or more FPGA's in the OBR unit is addedand/or modified in accordance with aspects of the invention. For thepurposes of this description, skilled individuals will understand thatthe various possible electronic data processing hardware, firmware, andsoftware combinations in the interrogator and “processors” may functionas a single electronic data processing unit, which may include the apriori information as described herein in a readable electronic memoryin one or more accessible locations in the interrogators/processors.

To define locations along fiber 1202, a curvilinear coordinate systems(t) is defined with an origin L₀ at fiber 1202's proximal end. Then,various shape sensing segment starts and ends are defined in usinglocations in the curvilinear reference frame to define one or more shapesensing segments 1206 (five are depicted, beginning at the proximal endand ending at the distal end of fiber 1202). As described above, thedefined segments may be immediately adjacent one another (see e.g., FIG.3A), or the defined segments may be spaced apart or overlap.

FIG. 12 also shows fiber shape measurement processor 1208 coupled tocommunicate (e.g., wired, wireless, optical coupling) with instrumentposition measurement processor 1211. As described above, fiber shapemeasurement processor 1208 receives strain information from each of thethree OBR 1204 units and outputs bend information as Δx, Δy, Δz, and{right arrow over (t)} for each of the segments 1206 in a signal 1210 toInstrument Position Measurement Processor 1211. In one illustrativeaspect, the bend information for each shape sensing segment is output asseven 16-bit words. The first six of the words each contain the fixedpoint representation of one of the Δx, Δy, Δz, t_(x), t_(y), and t_(z)values. A seventh word includes data counter, fiber identificationnumber, shape sensing segment identification number, and fault bits.Thus the position and orientation information for each shape sensingsegment is output to instrument position measurement processor 1211 in aformat that is easily subject to further processing. Instrument positionmeasurement processor 1211 also receives the instrument kinematic modelƒkin and the description of the geometrical constraints between thefiber and the kinematic chain of the instrument as described in Examples1-4. Instrument position measurement processor 1211 thus performs thenumerical optimization that leads to the estimate of the joint variables{right arrow over (q)} for the surgical instrument as described above inExamples 1-4 and equations 19-32. The surgical telerobotic motioncontroller 1212 then uses the joint position measurements to control thepose of the kinematic chain whose shape is being sensed by fiber 1202(e.g., by receiving inputs from a hand-operated controller andoutputting servomotor command signals that correspond to the receivedinputs). The operation of motion controller 1212 is for example asdescribed in, e.g., U.S. Pat. No. 6,493,608 (filed 7 Apr. 1999) and U.S.Pat. No. 6,424,885 (filed 13 Aug. 1999), and in U.S. Pat. Appl. No. US2007/0151389 (filed 20 Dec. 2006), all of which are incorporated hereinby reference, with the joint encoder input substituted with the jointposition measurements from controller 1212.

Aspects of the shape-sensing invention described herein include severaladvantages over known shape-sensing systems. Such advantages areespecially useful in aspects associated with telerobotically controlledsurgical instruments, since the necessary information is provided withthe precision needed for surgery.

First, the shape sensing is based on measurement of strain in the fiberalong each FBG with a very high resolution—on the order of microns forfibers of 10 m—as determined by the laser source coherence length. Sincethe FBG's are written adjacent one another in the core, the strainmeasurement has a very high resolution throughout the fiber length. Theresulting strain data set can be very large.

Second, the shape or state of the fiber is represented syntetically buteffectively for the purpose of determining the configuration of thekinematic chain embedding the fiber (e.g., of a surgical instrument orother device embedding the fiber). The state of the fiber is representedby the state of a set of segments defined in the fiber, and the segmentsmay be adjacent, separated, or overlapping. Each segment is defined by astart and end position along the fiber and has an associated start(base) and end frame of reference. The state of each segment of fiber isrepresented by the position and orientation of the end frame withrespect to the segment base frame.

Third, the position and orientation of the end frame of each segment iscomputed from the high resolution strain measurements on each core bymeans of an exact tridimensional integration over the segment, withintegration steps equal to the strain measurement resolution. Thetridimensional integration natively produces both the position andorientation of the end frame, i.e., a six degrees of freedommeasurement. The resulting full six degree of freedom Cartesian positionand orientation data facilitates further computation.

Fourth, the position and orientation of the segment end frame—thesegment data—is largely independent from the path taken by the length offiber due to the high resolution of strain data and exact 3Dintegration.

Fifth, the segment data can be merged with a priori information aboutthe kinematic chain (i.e., with the kinematic model of the embeddingstructure) to estimate the most likely position of one or more linksembedding the segment of fiber.

Sixth, the nature of the mechanical constraints between the fiber andthe embedding kinematic chain can be explicitly taken into account inthe process of merging the a priori information about the kinematicchain with the segment data in order to provide the most likely estimateof the link positions and orientations. In this way, the sliding,twisting, or curving of the fiber in its conduit does not negativelyaffect measurement of the kinematic chain's shape or pose.

1-9. (canceled)
 10. A shape sensing method comprising: defining alengthwise first segment in an optical fiber; defining a firstthree-dimensional reference frame at a start location of the firstsegment, wherein the first reference frame is a Cartesian frame, andwherein the first reference frame comprises a first axis that is normalto the fiber and a second axis that is tangent to the fiber; defining aCartesian base frame at an origin location in the fiber, wherein thebase frame comprises a first axis that is normal to the fiber and asecond axis that is tangent to the fiber, and wherein the first axis ofthe first reference frame and the first axis of the base frame intersecta common core in the fiber; interrogating the optical fiber for dataassociated with strain in the fiber; outputting first bend informationassociated with the first segment, wherein the first bend informationcomprises a Cartesian position at an end location of the first segmentwith respect to the first reference frame; and determining a shape ofthe fiber with respect to the base frame by using the first bendinformation, wherein determining the shape of the fiber furthercomprises using kinematic chain characteristics for a portion of thefiber in which no shape sensing segment is defined.
 11. A shape sensingmethod comprising: defining a lengthwise first segment in an opticalfiber, wherein the first segment is associated with a movable mechanicalconstraint between two links of a kinematic chain; defining a firstthree-dimensional reference frame at a start location of the firstsegment; interrogating the optical fiber for data associated with strainin the fiber; and, outputting first bend information associated with thefirst segment, wherein the first bend information comprises a Cartesianposition at an end location of the first segment with respect to thefirst reference frame.
 12. The method of claim 11: wherein the locationof the first segment start is within a first of the two links and alocation of the first segment end is within a second of the two linkssuch that the first bend information for the first segment is valid ifthe fiber slides in relation to the links as the mechanical constraintmoves.
 13. The method of claim 11: wherein the first bend informationfor the first segment is independent of the path of the fiber betweenthe two links.
 14. The method of claim 11: wherein the fiber is offsetfrom the centerline of the movable mechanical constraint.
 15. The methodof claim 11: wherein the first segment is associated with at least twomovable mechanical constraints between the two links.
 16. The method ofclaim 15: wherein the at least two movable mechanical constraints arecomponents of a flexible mechanism between the two links.
 17. The methodof claim 16: wherein outputting the first bend information comprisesminimizing an energy state of the two movable mechanical constraints.18. A method comprising: interrogating a segment of an optical fiberassociated with a kinematic chain to acquire information associated withbending strain in the segment; acquiring information associated with akinematic model of the kinematic chain from an electronic memory; andoutputting information associated with the kinematic configuration ofthe kinematic chain, wherein the information associated with thekinematic configuration is based on the acquired bending straininformation and the acquired kinematic model information.
 19. The methodof claim 18: wherein the kinematic chain is included in atelerobotically controlled surgical instrument.
 20. An apparatuscomprising: a kinematic chain comprising a proximal link, a distal link,and a joint coupled between the proximal and distal links; a shapesensing optical fiber associated with the kinematic chain, wherein ashape sensing segment in the optical fiber is defined in the fiberbetween the proximal link and the distal link, and wherein athree-dimensional reference frame is defined at a start location of theshape sensing segment; an interrogator that outputs strain informationassociated with shape sensing segment; and an electronic data processorthat receives the strain information and outputs a Cartesian position ofan end point of shape sensing segment with respect to the referenceframe.
 21. The apparatus of claim 20: wherein the reference framecomprises an x-axis, a y-axis, and a z-axis; and wherein the position isoutput as Δx, Δy, Δz.
 22. The apparatus of claim 20: wherein thereference frame is a Cartesian frame; and, wherein the reference framecomprises a first axis that is normal to the fiber and a second axisthat is tangent to the fiber.
 23. The apparatus of claim 22: wherein thereference frame comprises an x-axis, a y-axis, and a z-axis; and whereinthe position is output as Δx, Δy, Δz.
 24. The apparatus of claim 20:wherein the electronic data processor outputs a Cartesian orientation ofthe end point of the shape sensing segment with respect to the referenceframe.
 25. The apparatus of claim 24: wherein the orientation is outputas a tangent vector comprising direction cosines t_(x), t_(y), t_(z).